## AS Revision 2

I do not claim this is easy, and will provide good revision for C3 & C4 candidates

1   d ( 7x6 – 4x-3 )
dx

2   d ( 3x0.3 + 4x -0.4)
dx

3   d ( 3x⁴ – 4x) ³
dx

4   d ( 4 ( 23x – 4) ³ )
dx

5   d           4      .
dx    (5x – 3) ²

6   d ( sin 5x )
dx

7   d ( 6x⁵ )
dt

8   d ( 6t³ + 14y  )
dx

9   d  7(x6 – 4x) -3
dx

10  d (3x0.3 - 4x) -0.4
dx

11   d ( 3√x – 4   ³√x) 0.5
dx

12   d ( 4x-3) ( 23x³ – 4x)
dx

13   d            4 x     .
dx     (5x – 3)

14   d ( x sin x )
dx

15   d ( x³ sin x )
dx

16   d ( sin 3 x )
dx

17   d ( sin 3 x)-1     is also cosec 3x
dx

18   d ( e -0.4)
dx

19   d ( e 3x )
dx

20   d ( 4 e 3x - 2 )
dx

21   d ( 7x6 – 4e-3x )
dx

22   d ( 3e0.3x + 4e -0.4x)
dx

23   d ( 3x – 4ex) ³
dx

24   d    ( 23e-x – 4x) ³
dx

25   d     4x²(5x³– 3)-12
dx

26   d ( sin 5x³ )
dx

27   d ( ø e )
dø

28  d (ø e )
dø

Differentiation in context: finding the stationary values of a function

29  A box is built from a square sheet of side a. Square corners of side x are cut out and the result is folded and sealed.
a) Show that the box has volume V=(a-2x).(a-2x).x
b) By finding the stationary values of V, find the maximum volume of the box
c) Show that the maximum surface area for the box occurs when it is a cube.

30   A ball flies through the air according to the function
2U² y = xT – 5 x² (1+T²)  where U and T are constants for each flight.
a)  Find where dy/dx = 0, which is at the top of the flight, and check that this is half the range (the larger x-value that makes y=0).
b)  Treat 2U² y = xT – 5 x² (1+T²) as a quadratic in x and write the discriminant (the b² – 4ac part). For equal roots, the discriminant must be zero. Find the value of T when U=12 and y = 3.
c)  Treat 2U² y = xT – 5 x² (1+T²) as a quadratic in T and write the discriminant. For U=12 and y=3, find the x-value that gives equal roots.
d)  If y=0 and you consider x as a (product) function in T,  show that T = ±1 gives the maximum.

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